On Carvalho’s 𝐾-theoretic formulation of the cobordism invariance of the index

Moroianu, Sergiu

doi:10.1090/S0002-9939-06-08347-X

On Carvalho’s $K$-theoretic formulation of the cobordism invariance of the index
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Abstract:

We give an analytic proof of the fact that the index of an elliptic operator on the boundary of a compact manifold vanishes when the principal symbol comes from the restriction of a $K$-theory class from the interior. The proof uses non-commutative residues inside the calculus of cusp pseudodifferential operators of Melrose.

References

M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. MR 397797, DOI 10.1017/S0305004100049410
M. F. Atiyah and I. M. Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69 (1963), 422–433. MR 157392, DOI 10.1090/S0002-9904-1963-10957-X
M. F. Atiyah and I. M. Singer, The index of elliptic operators. I, Ann. of Math. (2) 87 (1968), 484–530. MR 236950, DOI 10.2307/1970715
Maxim Braverman, New proof of the cobordism invariance of the index, Proc. Amer. Math. Soc. 130 (2002), no. 4, 1095–1101. MR 1873784, DOI 10.1090/S0002-9939-01-06250-5
C. Carvalho, Pseudodifferential operators and applications to Index Theory on noncompact manifolds, Ph.D. thesis, Trinity College, University of Oxford (2003).
C. Carvalho, A $K$-theory proof of the cobordism invariance of the index, preprint math.KT/0408260 (2004).
Nigel Higson, A note on the cobordism invariance of the index, Topology 30 (1991), no. 3, 439–443. MR 1113688, DOI 10.1016/0040-9383(91)90024-X
Robert Lauter and Sergiu Moroianu, The index of cusp operators on manifolds with corners, Ann. Global Anal. Geom. 21 (2002), no. 1, 31–49. MR 1889248, DOI 10.1023/A:1014283604496
Robert Lauter and Sergiu Moroianu, An index formula on manifolds with fibered cusp ends, J. Geom. Anal. 15 (2005), no. 2, 261–283. MR 2152483, DOI 10.1007/BF02922196
Matthias Lesch, Deficiency indices for symmetric Dirac operators on manifolds with conic singularities, Topology 32 (1993), no. 3, 611–623. MR 1231967, DOI 10.1016/0040-9383(93)90012-K
Paul Loya, Tempered operators and the heat kernel and complex powers of elliptic pseudodifferential operators, Comm. Partial Differential Equations 26 (2001), no. 7-8, 1253–1321. MR 1855279, DOI 10.1081/PDE-100106134
Richard B. Melrose, The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics, vol. 4, A K Peters, Ltd., Wellesley, MA, 1993. MR 1348401, DOI 10.1016/0377-0257(93)80040-i
Richard B. Melrose, Pseudodifferential operators, corners and singular limits, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 217–234. MR 1159214
Richard B. Melrose, The eta invariant and families of pseudodifferential operators, Math. Res. Lett. 2 (1995), no. 5, 541–561. MR 1359962, DOI 10.4310/MRL.1995.v2.n5.a3
R. B. Melrose and V. Nistor, Homology of pseudodifferential operators I. Manifolds with boundary, preprint funct-an/9606005.
Richard B. Melrose and Paolo Piazza, Families of Dirac operators, boundaries and the $b$-calculus, J. Differential Geom. 46 (1997), no. 1, 99–180. MR 1472895
Sergiu Moroianu, $K$-theory of suspended pseudo-differential operators, $K$-Theory 28 (2003), no. 2, 167–181. MR 1995875, DOI 10.1023/A:1024537820594
Sergiu Moroianu, Cusp geometry and the cobordism invariance of the index, Adv. Math. 194 (2005), no. 2, 504–519. MR 2139923, DOI 10.1016/j.aim.2004.07.005
Liviu I. Nicolaescu, On the cobordism invariance of the index of Dirac operators, Proc. Amer. Math. Soc. 125 (1997), no. 9, 2797–2801. MR 1402879, DOI 10.1090/S0002-9939-97-03975-0
Boris Vaillant, Index- and spectral theory for manifolds with generalized fibred cusps, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 344, Universität Bonn, Mathematisches Institut, Bonn, 2001. Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 2001. MR 1933455